Connectedness versus diversification: two sides of the same coin

In the financial framework, the concepts of connectedness and diversification have been introduced and developed respectively in the context of systemic risk and portfolio theory. In this paper we propose a theoretical approach to bring to light the relation between connectedness and diversification. Starting from the respective axiomatic definitions, we prove that a class of proper measures of connectedness verifies, after a suitable functional transformation, the axiomatic requirements for a measure of diversification. The core idea of the paper is that connectedness and diversification are so deeply related that it is possible to pass from one concept to the other. In order to exploit such correspondence, we introduce a function, depending on the classical notion of rank of a matrix, that transforms a suitable proper measure of connectedness in a measure of diversification. We point out general properties of the proposed transformation function and apply it to a selection of measures of connectedness, such as the well-known Variance Inflation Factor.


Introduction
The idea that risk reduction for an investment portfolio can be achieved through a diversification approach is central in the Markowitz model, see Markowitz [19], and one of the principal reasons of its popularity. There is a wide consensus on the risk reduction properties of the diversification, both from practitioners and academics. The qualitative definition of diversification is natural for portfolio managers: an investment portfolio is well diversified when it is not exposed to individual shocks occurring to its constituents. The formal definition of diversification remains an elusive concept and it is hardly made explicit in portfolio optimization studies, with an exception for the paper by Meucci [20] and related studies. In its paper, Meucci proposes to use principal component analysis to extract uncorrelated risk factors starting from the underlying assets.
The vagueness of its definition gave rise to a huge literature focusing on different aspects of diversification. Many different approaches for diversification based portfolio strategies have been studied in the literature: among the others, we recall the naive diversification strategy and the measurement of its out of sample performance, see DeMiguel et al. [7], the equally risk contribution diversification strategy, see Roncalli and Weisang [23] and Qian [22], the application of the approach proposed by Meucci, see Lohre et al. [17], the determination of maximum diversification portfolios, see Choueifaty and Coignard [5], the investment strategy based on the maximization of the diversification ratio, see Choueifaty et al. [6], the strategy based on the generalized Rao's entropy used as a diversification measure, see Choueifaty et al. [6]. Moreover, the literature presents further approaches that relate the diversification to different aspects of portfolio composition, like sector diversification and geographical diversification, see for example Hauser and Vermeersch [14] and Diamond and Abdullah [8], or product based diversification, see for example Bernardi et al. [1], for the diversification approach based on commodities risk factors. In this paper we refer to the axiomatic definition of coherent diversification measures proposed by Koumou and Dionne [16].
While the idea of diversification approximately dates back to 1950s, the concept of connectedness is relatively new. It has been first introduced by Diebold and Yilmaz, see Diebold and Yilmaz [9], and successively developed in subsequent papers, see among the others Diebold and Yilmaz [10] and Diebold and Yilmaz [11], in which the authors well describe the concept of connectedness, pointing out the lack of a formal definition and the resulting vagueness of the notion. Recently, an attempt to formalize the concept of connectedness through an axiomatic approach has been proposed in Maggi et al. [18]. The idea behind connectedness is to measure the degree of the inter-relations and the inter-dependencies between the components of a whole system and to summarize the information in a single number. Despite its potential generality, restricting to the financial context, a high connectedness reflects a strong interdependence between the elements of the system. This interpretation highlights the intuitive relation between connectedness and systemic risk. For a comprehensive review of the measures proposed in the literature for systemic risk analysis we refer to the survey by Bisias et al. [4].
In the context of portfolio theory, diversification is usually achieved through the computation of the optimal weights to invest in. On the opposite, connectedness is not directly related to portfolio theory and portfolio weights are not considered when evaluating the connectedness of a market or an economic system or even the assets of a portfolio. Despite the differences and the distinctive features among connectedness and diversification, in this paper we aim at investigate and highlight the deep relation between the two concepts.
An important similarity can be found in the role of correlation. As pointed out in Diebold and Yilmaz [11], in the literature on connectedness the use of correlation-based measures is widespread and the concept of connectedness may be seen as a generalization of the concept of correlation while, on the other hand, correlation is the standard base for the evaluation of diversification. One further link between connectedness and diversification can be found in the role of the eigenvalues in both fields, see for instance Meucci [20] and Maggi et al. [18], where the eigenvalues are used to evaluate both diversification and connectedness.
In accordance to standard portfolio theory asserting that the portfolio risk can be decomposed into its systematic and idiosyncratic components, diversification is geared at eliminating the idiosyncratic risk while connectedness estimates the systemic risk component. In this framework, the relation between the two concepts is immediate and connectedness can be thought as the complement of the diversification with respect to the the total risk of a portfolio. Referring to the idiosyncratic and systemic components of the risk, it is useful to interpret the monotonicity properties of the two measures with respect to the size of the portfolio; in particular, diversification improves with large portfolios while connectedness, representing the systemic component, is not positively affected by a growing number of assets. We will show that a high level of connectedness corresponds to a reduced possibility to build a well diversified portfolio. On the opposite, the possibility to build a well diversified portfolio is related to a low level of connectedness between the assets, i.e. the assets provide good opportunities of diversification.
In this paper, in order to highlight the similarities between connectedness and diversification, we explicitly introduce the weights in the computation of connectedness, generalizing its definition and showing that standard measures of connectedness can be related to the special case of the equally weighted portfolios. Further, we effectively analyze this correspondence by comparing the axiomatic frameworks of the Proper Measures of Connectedness (PMCs) and the Coherent Portfolio Diversification Measures (CPDMs), see Maggi et al. [18] and Koumou and Dionne [16] respectively. We note that, due to the peculiarities of the two settings, the requirements for PMCs are less restrictive than the ones for CPDMs. As a consequence, a specific PMC needs to verify some additional properties to be properly related to a CPDM.
Our approach is general. We introduce a suitable function F and prove that, if applied to a subclass of PMCs verifying given additional properties, such as, for example, quasi-convexity and homogeneity, it transforms such measures into CPDMs. Then, we consider a selection of four PMCs, the Maximum Variance Inflation Factor (M-VIF), see Belsey et al. [2], the Power Mean measures μ 1 k , see Maggi et al. [18], the Market Rank Indicator, see Figini et al. [12] and the Cumulative Risk Fraction h k , see Billio et al. [3], and we apply F defining four original diversification measures. We prove in details that F M-VIF is a CPDM and show that the measures of diversification F μ 1 k , F s k and F h k partially verify the theoretical requirements of CPDMs.
The paper is organized as follows: in Sect. 2 we recall the axiomatic frameworks of PMCs and CPDMs; in Sect. 3 we introduce the function F providing the relationship between the concepts of connectedness and diversification, dwelling on the additional required properties on PMCs necessary to construct a CPDM; in Sect. 4 we apply F to a selection of PMCs and prove the properties they verify as induced measures of diversification; finally, in Sect. 5 we discuss the proposed results.

Connectedness and diversification: axiomatic definitions
In this section we recall the general notions of Proper Measures of Connectedness (PMCs) and Coherent Diversification Measures (CDMs) briefly specifying the economic interpretation of the theoretical properties required for the axiomatic definition of the two families of measures; for a more detailed description we refer to Maggi et al. [18] and Koumou and Dionne [16] respectively. In order to compare the measures of connectedness and diversification in a meaningful perspective, we refer to the portfolio allocation scheme, where the CPDMs have been principally developed.

Notation and preliminaries
Let m ≥ n ≥ 2, let Mat m×n be the set of m × n real matrices and M m×n be the subset of Mat m×n containing all the full-rank matrices, i.e. rank(A) = n, ∀A ∈ M m×n . We indicate with 0 m×n and 1 m×n the m × n matrices whose elements are all equal to zero and one respectively. For brevity, we will also use the simpler notation 1 n , or even 1 if the size of the vector is clear from the context, to denote the vector 1 n×1 . Throughout the paper, we will interpret the entries of the column vectors A j , with j = 1, . . . , n, of any matrix A = (a i j ) as the historical observations of the jth return of the portfolio, so that the m × n matrix A is the usual matrix of portfolio returns with m historical data and n assets. We denote by A j the mean of A j and let A = (A 1 1 m×1 , . . . , A n 1 m×1 ); A contains the averages of the historical returns of the assets in the portfolio. We let A t be the transposed of A and σ 1 (A) ≥ . . . ≥ σ r (A) > 0, with r = rank(A), be the singular values of A listed as usual in nonincreasing order; further, for any α ∈ R, we denote by A + α and α A the m × n matrices whose (i, j) entry is a i j + α and αa i j , respectively. We let P = ( p 1 , . . . , p r ) be the list of pivots obtained by the classical Gauss-Jordan elimination applied to A and A P = (A p 1 , . . . , A p r ) ∈ M m×r . We denote by ρ(·, ·) the Pearson correlation coefficient, by ·, · the standard scalar product and by · the Euclidean norm of R n . Finally, we let W n = {w = (w 1 , . . . , w n ) t ∈ R n ≥0 | n j=1 w j = 1} be the set of long-only portfolios associated to A, so that w j is the weight of asset A j in the portfolio w, and we define W = diag(w) ∈ Mat n×n as the diagonal matrix with elements w 1 , . . . , w n . A portfolio made up of a single asset j is denoted by e t j , where e 1 , . . . , e n is the standard basis of R n .
We denote by H m,n the set of all positive real-valued functions defined over the set of matrices Mat m×n \ {0 m×n } and by H m,n w the set of all positive real-valued functions defined over W n × Mat m×n \ {0 m×n }. In the following definition we list some properties of functions belonging to H m,n .

Definition 1
Let g ∈ H m,n ; then g is said to be Mat m×n having no rows and no columns equal to 1 1×n and 1 m×1 respectively, and

Remark 1
It is immediate to observe that the invariance under one scalar multiplication is a very strong property which implies that the function is invariant under (even different) simultaneous scalar multiplication of each column. Further, in particular, it implies the degree 0 homogeneity property.

Connectedness
Definition 2 (PMCs) A real-valued function C : M m×n → R, defined by C(A) for each A ∈ M m×n is a Proper Measure of Connectedness (PMC) if it satisfies the following minimal properties C1, C2, C3 and C4. a 3 )) where, with a small abuse of notation, we assume that (a 1 , It is useful to observe that if C satisfies property (v) of Definition 1, then C satisfies property C3.
In the financial context, the minimal properties required in Definition 2 for a PMC are intuitive. If we exclude property C1 that is a pure technical condition, a PMC is not affected by the order the assets are considered, see property C2. Property C3 requires that a positive rescaling in the data does not significantly impact the structure of connectedness. Finally, with property C4, a PMC is required to embody at least the information of the correlation structure of the data so that a higher correlation results in a higher connectedness. The opposite implication does not hold; an increase in the value of connectedness could depend on different causes than an increase in the value of correlation among the data.
Definition 2 provides the general notion of PMCs, as given in (Maggi et al., [18]). In this paper, in order to use PMCs in the context of standard portfolio theory, we will consider connectedness measures defined on matrices AW , where W = diag(w) is the diagonal matrix associated to a tuple of portfolio weights w ∈ W n . In this way, Definition 2 boils down to the standard definition of PMCs when the function C is homogeneous of degree 0 and the considered portfolio is 1 n 1, that is the equally weighted portfolio.

Diversification
In the following definition we recall the Coherent Portfolio Diversification Measures (CPDMs).

Definition 3 (CPDMs) A real-valued function
(Reverse risk degeneracy) For any w ∈ W n such that w = e j , j = 1, . . . , n, and w i > 0, i = 1, . . . , n, we consider the equation Φ(w, A) = Φ in the variable A ∈ Mat m×n and assume that a solution A * exists. Then A * is lower comonotonic.
for each w ∈ W n .
In the financial context, the minimal properties for a CPDM listed in Definition 3 have the following economic interpretation. The preference towards diversification such that holding different assets increases diversification is translated by property D1. Property D2, size degeneracy, reflects that single asset portfolios are equivalent to the minimum level of diversification. Properties D3 and D4, risk degeneracy and reverse risk degeneracy, respectively state that including perfect similar assets in a portfolio do not help diversification and that a portfolio composed by assets with independent returns and single asset portfolios do not have the same level of diversification. Property D5, duplication invariance, is required to avoid that diversification measurement is affected by multiple representative assets. Property D6, size monotonicity, reflects the relation between portfolio diversification and the number of assets in the portfolio. Adding a deterministic cash return does not impact portfolio diversification: this idea is described by property D7. Diversification is not affected by scalar transformations of the input data, see property D8 about homogeneity. Finally, as in the context of connectedness, diversification is required to be independent from any permutation of the assets in a portfolio, see property D9 about symmetry.

From connectedness to diversification
In this section we propose a method to construct original portfolio diversification measures starting from given PMCs. In order to explicitly define a subclass of CPDMs we need to require the PMCs to verify some extra properties, restricting the class of PMCs that are of potential interest. The passage from connectedness to diversification is possible through the use of a specific transformation function F : H m,n → H m,n w defined as follows.
Definition 4 (Transformation function) Let C ∈ H m,n and f C ∈ H m,n w be the function defined by: The transformation function F : H m,n → H m,n w is defined by F(C) = F C ∈ H m,n w , for each C ∈ H m,n , where F C acts on W n × Mat m×n \ {0 m×n } as follows: for each w ∈ W n and A ∈ Mat m×n \ {0 m×n }.
Note that in the rest of the paper we will only consider the nontrivial and meaningful case rank(AW ) > 0. Intuitively, the behaviour of F C agrees with the distinguishing feature of diversification: indeed, if we consider the simplest case of the equally weighted portfolios, F C decreases as the function C increases, which highlights the main relationship between the notions of diversification and connectedness (if the overall connectedness between the assets increases, the diversification opportunities when creating a portfolio with these assets are expected to reduce).
The following proposition gathers some basic properties on F C . [1, +∞). For each nonzero A, A 1 , A 2 ∈ Mat m×n and w, w 1 ,

Proposition 1 Let C be a function of H m,n that assumes values in
∈ Mat n×n , the following properties hold.
which yields rank(AW ) = 1 and consequently C(AW ) = 1. On the other hand, imposing the positivity of the function F C , the condition rank(AW ) = 1 implies that C(AW ) = 1, and so F C (w, A) = 0, so item 1 is proved. 2. From now on, we consider the case rank(AW ) ≥ 2. Following the hypothesis on the function C we have f C (w, A) > 1, therefore item 2 is proved. 3. Using inequality (1) and Definition 4, item 3 immediately follows. 4. The hypothesis of item 4 yields rank(A 1 W 1 ) ≤ rank(A 2 W 2 ) − 1 so, using item 3, we have: therefore item 4 is proved. We now provide a deeper characterization of F C on the base of additional properties required for C. To this end, we need to introduce the following technical lemma yielding useful features of the rank operator in order to investigate the concavity properties of F C . Lemma 1 Let A ∈ M m×n and w 1 , w 2 ∈ W n , with W 1 = diag(w 1 ), W 2 = diag(w 2 ) ∈ Mat n×n . For any α ∈ (0, 1), the following result holds: rank(AW 1 ) = rank(W 1 ) and rank(AW 2 ) = rank(W 2 ). Further, W α is a diagonal matrix whose entries are the strictly convex combinations of the corresponding elements of W 1 and W 2 , so that its strictly positive entries are indexed by I 1 ∪ I 2 . It easily follows that with strict inequality if and only if I k I 1 ∪ I 2 , for each k = 1, 2. Therefore, combining (2) and (3), the result follows.

Remark 2
The result of Lemma 1 can be easily transformed in terms of concavity by simply extending it to the case α ∈ [0, 1]. Indeed, using the notation of Lemma 1, for any α ∈ [0, 1] the rank operator satisfies: with strict inequality for some value of α ∈ (0, 1) if and only if I k I 1 ∪ I 2 , k = 1, 2. We conclude that the rank operator is a is a quasi-concave function.
On the base of the previous results, the following proposition highlights important concavity properties of the function F C . [1, +∞) and satisfying property (vi) of Definition 1. Then, F C is a quasi-concave function of H m,n , that is for each nonzero A ∈ Mat m×n , α ∈ [0, 1], w 1 , w 2 ∈ W n , the following result holds:
Proof Let W = diag(w) ∈ Mat n×n and W + = diag(w + ) ∈ Mat (n+1)×(n+1) . Recalling Definition 4, it is enough to prove that rank(A + W + ) = rank(AW ) and C(A + W + ) = C(AW ). The first equality is immediate; we prove the second equality by considering the following three cases: w + k = w k and w + n+1 = 0; w + k = 0 and w + n+1 = w k ; w + k > 0 and w + n+1 > 0. In the first case, w + k = w k and w + n+1 = 0, the matrix A + W + can be expressed as A + W + = (AW |0 m×1 ), so the invariance by size property of C yields C(A + W + ) = C(AW ). In the second case, w + k = 0 and w + n+1 = w k , the matrix A + W + can be expressed as A + W + = (AW |0 m×1 )Π, where Π is a permutation matrix only exchanging the kth and (n + 1)th positions. Applying the invariance by columns permutations and by size we obtain C(A + W + ) = C(AW ). In the last case, w + k > 0 and w + n+1 > 0, with w + k + w + n+1 = w k , we let w k + = bw k , with b > 0. The matrix A + W + can be expressed as (A 1 w 1 , . . . , A k−1 w k−1 , α A k w k , A k+1 w k+1 , . . . , A n w n , A k w + n+1 ), and applying the invariance for size and under one scalar multiplication we obtain C(A + W + ) = C (A 1 w 1 , . . . , c A k w k , . . . , A n w n ) = C(AW ).

Proposition 5 (Translation invariance) Let C be a function of H m,n that satisfies property (i) of Definition 1, let A ∈ Mat m×n be a nonzero matrix and α ∈ R.
If no rows and no columns of A are equal to 1 1×n and 1 m×1 respectively, then Proof Let W = diag(w) ∈ Mat n×n and note that the matrix A + α can be expressed as Proof Let W = diag(w) ∈ Mat n×n . By recalling Definition 4 and exploiting the equality rank(α AW ) = rank(AW ) and the degree 0 homogeneity of C, it immediately follows that F C (w, α A) = F C (w, A).

Proposition 7 (Symmetry) Let C be a function of H m,n that satisfies property (iii) of Definition 1, et A ∈ Mat m×n be nonzero and Π ∈ Mat n×n be a permutation matrix. Then
We consider the matrix AΠ W Π = AΠΠ t W Π = AW Π: since rank(AW Π) = rank(AW ) and, by hypothesis on C, C(AW Π) = C(AW ), recalling Definition 4, the result immediately follows.

Induced portfolio diversification measures
In this section, starting from four fixed PMCs, the Maximum Variance Inflation Factor (M-VIF), see Belsey et al. [2], the Power Mean measures μ 1 k , see Maggi et al. [18], the Market Rank Indicator (MRI), see Figini et al. [12], and the Cumulative Risk Fraction h k , see Billio et al. [3], and using the function F introduced in Sect. 3, see Definition 4, we explicitly construct original portfolio diversification measures. We prove that F M-VIF is a CPDM, whereas F μ 1 k , F s k and F h k only partially verify the theoretical requirements of CPDMs. Since, in general, the PMCs are only defined in the case of full-rank matrices, we introduce for each considered PMC a possible generalization that, exploiting the classical Gauss-Jordan elimination method, allows us to treat even the case of rank-deficient matrices. To this end, in the rest of the section, for each nonzero matrix A ∈ Mat m×n we denote by r = rank(A − A) and by (A − A) P = ((A − A) p 1 , . . . , (A − A) p r ) the matrix obtained applying the Gauss-Jordan elimination method to A − A.

The Variance Inflation Factor
In Maggi et al. [18], based on the classical econometric measure known as Variance Inflation Factors (VIFs), see for instance Belsey et al. [2], it is proved that the maximum VIFs provides a measure of connectedness. In the following definition we recall and extend this notion to be defined for mean-centered variables even in the case of rank deficient matrices (we also refer to Gross [13] for the definition of the centered VIFs).  Proof The proof deeply exploits the properties of M-VIF shown in Proposition 8. In the following we prove that F M-VIF satisfies the following axioms.

The Power Mean measures 1 k
The Power Mean measures, see Maggi et al. [18], are a family of proper measures of connectedness defined as the ratio between the greatest singular value of the returns and a specified power mean of its smallest singular values. In the following definition we recall this notion in the special case of arithmetic mean, and extend it to be defined for mean-centered variables even in the case of rank-deficient matrices.

Definition 6 Let
for each A ∈ Mat m×n \ {0 m×n }, where M 1 denotes the arithmetic mean. Proof Let A ∈ Mat m×n be a nonzero matrix and σ j ((A − A) P ) be the singular values of (A − A) P . Using the internal property of the arithmetic mean, we get which yields μ 1 k (A) ≥ 1, for each k = 1, . . . , r . We prove properties (i), (ii), (iii), (v), (vi).
(i) We consider the case that A has no rows and no columns equal to 1 1×n and 1 m×1 respectively and let B = (b i j ) ∈ Mat m×n such that b i j = β j ∈ R for each i = 1, . . . , m and j = 1, . . . , n. It is straightforward to verify that the mean of (A + B) j is A j + β j so that, if we denote by A + B the matrix of the column means of A + B, we have (v) The property follows from the same property on the measure of connectedness μ 1 k , see Maggi et al. [18]. (vi) We consider A, B ∈ M m×n . A result of convex analysis, see for instance Qi and Womersley [21], states that the largest singular value and the sum of the k smallest singular values, viewed as matrix functions, are convex and concave functions respectively. Thus, for each α ∈ [0, 1], we get: It is straightforward to verify that max α∈[0,1] F(α) = max{μ 1 k (A), μ 1 k (B)}, which yields the quasi-convexity property μ 1
Proof The result exploits Proposition 12 and is obtained arguing analogously to the proof of Proposition 9 to which the reader is referred.

The Cumulative Risk Fraction
The notion of Cumulative Risk Fraction, see Billio et al. [3], is defined as the portion of the variability of the returns explained by the first principal components. In Maggi et al. [18] it is proved that the Cumulative Risk Fraction provides a measure of connectedness. In the following definition we recall this notion, rescale it to assume values in the range [1, +∞) and extend it to be defined even in the case of rank deficient matrices. .

Definition 8 Let
for each A ∈ Mat m×n \ {0 m×n }.
It is easy to observe that, in the case of full-rank matrices, Definition 8 coincides with the inverse of the Cumulative Risk Fraction as given in Billio et al. [3]. Proof Using the definition, it is immediate to verify that h k (A) ≥ 1, for each k = 1, . . . , r . Further, since the Cumulative Risk Fraction is homogeneous of degree 0, see Maggi et al. [18], the same property holds for the measure h k . For the proof of properties (i), (ii), (iii), (v) we refer to the analogous case of μ 1 k (see proof of Proposition 10).
Proof The proof is based on the results of Proposition 14 and similar to the proof of Proposition 9, to which the reader is referred.

Conclusions
In this paper we compared the PMCs and the CPDMs starting from their respective axiomatic characterizations. The generality of the proposed approach permits to discuss the properties a PMC needs to verify in order to belong to the class of CPDMs, once the transformation defined by the function F is applied. Such transformation function F supports the intuition described in the introduction that a strong relation links the concepts of connectedness and diversification; in particular, given the number of assets in a portfolio, an increase in the connectedness reflects a reduction of the diversification opportunities. The central result of the paper is the definition of the induced CPDM starting from a PMC that verifies some required additional properties. In practice, we showed that the VIF induces a CPDM, while other popular PMCs, as the Cumulative Risk Fraction, the Market Rank Indicator and the Power Mean measures, including the special case of the condition number, only partially verify the requirements to define an induced CPDM. While in the present paper, for the sake of simplicity, we focused on the transformation from connectedness to diversification, in our future research we plan to investigate the opposite transition, from diversification to connectedness.

Conflict of interest
The authors declare that they have no conflict of interest.
Funding Open Access funding provided by Università degli Studi di Genova.
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